Translating graphs

What is a translation?

A translation is basically when we move a graph (without changing its shape).

We can represent a translation using a vector of coordinates, where the top number represents the movement in the direction, and the bottom number represents the movement in the direction.

For example, the vector means we move 3 units to the right (positive direction) and 2 units down (negative direction).

Equation of a translated graph

If we have a function , and we translate it using the vector , the equation of the translated graph will be:

This means we subtract from inside the function, and add to the whole function.

The way to remember this is:

• Inside the bracket is , and it’s the opposite of what the vector says
• Outside the bracket is the whole function, and it’s the same as what the vector says

There’s an easier way to think of this, though:

• If we want to move the graph in the direction, we subtract from the (which happens to be inside the function).
• If we want to move the graph in the direction, we subtract from the .

Using this, a translation of would give us , which rearranges to the same equation as above: .

Example: find the equation of the graph translated by

• Start with the originaal equation:
• Translate using the vector :
• Simplify:
• Answer: .